In this article, we define an information-theoretic entropy based on the Ihara zeta function of a graph which is called the Ihara entropy. A dynamical system consists of a billiard ball and a set of reflectors correspond to a combinatorial graph. The reflectors are represented by the vertices of the graph. Movement of the billiard ball between two reflectors is represented by the edges. The prime cycles of this graph generate the bi-infinite sequences of the corresponding symbolic dynamical system. The number of different prime cycles of a given length can be expressed in terms of the adjacency matrix of the oriented line graph. It also constructs the formal power series expansion of Ihara zeta function. Therefore, the Ihara entropy has a deep connection with the dynamical system of billiards. As an information-theoretic entropy, it fulfils the generalized Shannon-Khinchin axioms. It is a weakly decomposable entropy whose composition law is given by the Lazard formal group law.

中文翻译：

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台球系统及其在信息论熵中的应用

在本文中，我们基于图的 Ihara zeta 函数定义了一个信息论熵，称为 Ihara 熵。一个动力系统由一个台球和一组对应于组合图的反射器组成。反射器由图形的顶点表示。台球在两个反射器之间的运动由边缘表示。该图的素循环生成相应符号动力系统的双无限序列。给定长度的不同质数循环的数量可以用有向线图的邻接矩阵表示。它还构造了 Ihara zeta 函数的形式幂级数展开式。因此，井原熵与台球的动力系统有着很深的联系。作为信息论的熵，它满足广义的香农-欣钦公理。它是一个弱可分解熵，其组成定律由 Lazard 形式群定律给出。